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Results tagged with graph-theory
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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
65
votes
Strengthening the induction hypothesis
Here is a bit of advice that took me a while to learn:
You don't need to know what you are proving when you start to write a proof by induction.
The following method isn't needed for easy prob …
25
votes
Accepted
Can you determine whether a graph is the 1-skeleton of a polytope?
A few comments:
In general, you can't tell the dimension of a polytope from its graph. For any $n \geq 6$, the complete graph $K_n$ is the edge graph of both a $4$-dimensional and a $5$-dimensional po …
- 156k
24
votes
Rock-paper-scissors...
Your sequence is Sloane A096368. Sloane links to this page, which has files of all examples up to $13$ vertices. MathSciNet has 30 papers with "regular tournament" in the title, none of which seem to …
- 156k
15
votes
Accepted
How can I prove that a particular family of graphs is integral?
$\def\CC{\mathbb{C}}$The specturm is integral.
The following trick is very useful in computing spectra of highly symmetric graphs. Let $G$ be a finite graph, let $\Gamma$ be a group of symmetries of …
- 156k
14
votes
What is the Tutte polynomial encoding?
I'm going to take the opportunity to advertise one of my results: A number of people have said to me "I always thought there was something $K$-theoretic about the Tutte polynomial". Alex Fink and I ha …
- 156k
11
votes
What is the cycle structure of a graph?
This is a vague question, but here is an attempt at an answer. Let $G$ be a graph, let $E$ be the set of edges of $G$, and let $C \subset 2^E$ be the set of cycles of $G$. Then knowing $C$ is equivale …
- 156k
11
votes
Accepted
Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys?
Regarding (2), the answer is still "no". The following counter-example is from:
Tamura, Akihisa Transformation from
arbitrary matchings to stable
matchings J. Combin. Theory Ser. A
62 (1993) …
- 156k
11
votes
Accepted
Number of spanning forests in a graph
Two points:
(1) There is a result which is very similar to this. Let $G$ be a graph. Let $A$ be the matrix whose rows and columns are indexed by the vertices of $G$, where $A_{ij}$ is negative the nu …
- 156k
10
votes
Accepted
Lower bound on # spanning trees in a connected graph
A quick google turns up:
Undirected simple connected graphs with minimum number of spanning trees by Zbigniew R. Bogdanowicz. According to this paper, the optimal graph is built as follows: Start wit …
- 156k
10
votes
Connectivity of the Erdős–Rényi random graph
Here is an easy proof for $C>2$, and a fairly easy proof for $C>1$.
Define a cut of a graph $G$ to be a partition of the vertices of $G$ into two sets which are crossed by no edges. So a graph has a …
- 156k
9
votes
Accepted
Where is it shown how to construct a decomposition tree for a series-parallel graph in linea...
It is easy enough to build the tree from a definition 2 description. Here is a sketch:
Suppose that SP graph $G$ occurs from subdividing graph $G'$ at an edge $e$, either in series or in parallel. Re …
- 156k
9
votes
Accepted
Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
I suspect that the optimum, for a cube of side length $2$, is $2^k \sqrt{3} - 2 \sqrt{3}+2$. Note that the optimum if we use edges of the cube is $2 (2^k-1)$, so this is better by a factor of roughly …
- 156k
8
votes
A question on representation of graphs
I have now tried a few different strategies which are all winding up at $d \approx n \log_2 n$. I'll describe the simplest of these. (David Epstein's strategy with the spanning tree gives $d \approx n …
- 156k
8
votes
Complexity of determining if two graphs have same cycle matroid?
I believe that they have the same complexity, but writing up the details has proved painful and it is possible I am missing something. Let me give the first part of my answer, and see whether you actu …
- 156k
7
votes
Does this graph exist?
What if you take $5$ paths with $k$ vertices each and glue them together at the endpoints? So $5(k-2)+2=5k-7$ vertices in all, and circumference $2(k-1)=2k-2$.
- 156k